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#### charlottewill

##### New member

- Mar 7, 2012

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We've been asked to solve this system of equations

dW/dt = = αW−βV

dV dt = −γV+δW

by proposing that

W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt).

so I've differentiated both W(t) and V(t) to give W'(t) = ωA cos(ωt) - ωB sin(wt) and V'(t) = ωC cos(ωt) - ωD sin(wt) and replaced with the dW/dt and dV/dt in the system to give the expressions

ωA cos(ωt) - ωB sin(wt) = αW−βV

ωC cos(ωt) - ωD sin(wt) = −γV+δW

and now I've replaced the W and V's on the RHS with W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt) (originally given in the question) and I've factored out the constant and the parameters of cos(ωt) and sin (ωt) to give two simultaneous equations equal to zero which are

(ωA-αB+βD) cos(ωt) + (βC-αA-ωB) sin(ωt) = 0

(ωC+γD-δB) cos(ωt) + (γC-ωD-δA) sin(ωt) = 0

So how do I find the value of the constants A,B,C and D in terms of the parameters α,β,γ,δ ω?

Please can somebody help me

Thanks,

Charlotte

dW/dt = = αW−βV

dV dt = −γV+δW

by proposing that

W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt).

so I've differentiated both W(t) and V(t) to give W'(t) = ωA cos(ωt) - ωB sin(wt) and V'(t) = ωC cos(ωt) - ωD sin(wt) and replaced with the dW/dt and dV/dt in the system to give the expressions

ωA cos(ωt) - ωB sin(wt) = αW−βV

ωC cos(ωt) - ωD sin(wt) = −γV+δW

and now I've replaced the W and V's on the RHS with W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt) (originally given in the question) and I've factored out the constant and the parameters of cos(ωt) and sin (ωt) to give two simultaneous equations equal to zero which are

(ωA-αB+βD) cos(ωt) + (βC-αA-ωB) sin(ωt) = 0

(ωC+γD-δB) cos(ωt) + (γC-ωD-δA) sin(ωt) = 0

So how do I find the value of the constants A,B,C and D in terms of the parameters α,β,γ,δ ω?

Please can somebody help me

Thanks,

Charlotte

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